# Geometry Visualized voiceover, direct four-part version

This is a local one-off review artifact. It is not autonomous sandbox evidence.

## Part 1: Rigid transformations

Geometry begins with invariant properties: measurements that remain unchanged when shapes move. In Euclidean space, translating, rotating, or reflecting a shape changes its position while preserving its distances and angles.

## Part 2: Unit circle and sine wave

The unit circle connects rotation with trigonometry. As a point travels around the circle, its horizontal and vertical positions change over time. Following the vertical position draws a sine wave.

## Part 3: Area rearrangement

Rearranging these four right triangles reveals the Pythagorean theorem. The same outer square first leaves open areas a squared and b squared, then leaves open area c squared. Since the triangles have not changed, a squared plus b squared equals c squared.

## Part 4: Curved geometry

Curved surfaces follow different geometric rules. On the sphere, this geodesic triangle has an angle sum greater than one hundred eighty degrees. On the saddle, geodesic paths diverge, and the triangle's angles sum to less than one hundred eighty degrees.